Introduction. We discuss here a problem raised by a remark due to Skolem, appearing in [Sko], p. 23, Bemerkung 1, concerning the integer so- lutions of the equation

LXXVIII.2 (1996)

Remarks on a question of Skolem about the integer solutions of x 1 x 2 − x 3 x 4 = 1

by

Umberto Zannier (Venezia)

Introduction. We discuss here a problem raised by a remark due to Skolem, appearing in [Sko], p. 23, Bemerkung 1, concerning the integer so- lutions of the equation

(1) x ₁ x ₂ − x ₃ x ₄ = 1.

He pointed out that it seemed unlikely that all the integral solutions could be obtained from a fixed polynomial parametrization of (1) by letting the variables run through all integers (actually Skolem considered the more gen- eral equation det(x ij ) = 1, 1 ≤ i, j ≤ n). Observe (cf. [Sz]) that we can find infinitely many polynomial solutions of (1) with coefficients in Z by consider- ing the generic continued fractions a ₀ + _a ¹

1

+ . . . _a ¹

n

where the a _i are variables.

As is well known, we can write the value of such fraction as p n /q n , where p _n , q _n are polynomials in the a _i ’s satisfying p _n+1 q _n − q _n+1 p _n = (−1) ⁿ . How- ever, no such formula produces all integral solutions of x ₁ x ₂ − x ₃ x ₄ = 1 by letting the a i run through all integers: in fact, it can be easily shown by in- duction on n that, if the a _i ∈ Z, the rational number p _n /q _n has a continued fraction expansion with a number of terms bounded by a function of n only.

(These references and remarks were pointed out to me by A. Schinzel.) In the present note we shall prove Skolem’s expectation under the as- sumption (apparently very strong, but see Remarks 3, 4 and Example 1 below) that the polynomials appearing in the parametrization depend on three variables only (i.e. like the dimension of the quadric defined by (1));

actually, we shall argue over any number field k, with ring of integers O _k . We have the following

Theorem 1. Given a finite number of polynomial solutions of (1) x _i = p ^(j) _i ∈ k[t ₁ , t ₂ , t ₃ ], j = 1, . . . , h, there exist numerical solutions of (1), (a 1 , a 2 , a 3 , a 4 ) ∈ O ⁴ _k which cannot be obtained from any of the polynomial ones by specializations of the t _i ’s to integers in O _k .

[153]

We shall deduce Theorem 1 from Theorem 2 below, which leads to similar statements for more general varieties.

With no restriction on the number of variables in the parametrization, we have strong evidence that the result should not be true for general fields k;

more precisely, relating the question with Artin’s conjecture on primitive roots, we shall point out in Remark 3 how the truth of the Generalized Rie- mann Hypothesis implies the existence of counterexamples to the analogue for Z[ √

2] of Skolem’s belief, already allowing five variables. (The method actually works for any k which is neither the rational field nor an imaginary quadratic field.)

This fact also says that no refinement of our method, which works for rings of integers of any number field k, can possibly provide a complete answer to Skolem’s feeling, which, if correct, seems to depend on peculiar properties of Z compared to general O _k (like e.g. the structure of units).

Actually, it is possible that the example of Remark 3, for the field Q( √ 2) and five variables, gives as a byproduct an example over the rational field, in ten variables. We shall briefly explain this in Remark 4, pointing out the connection of the problem with a simply stated general question on polynomial ideals.

On the other hand, it would be interesting to sharpen the theorem by allowing the polynomials to depend on four variables. Of course this would be conjecturally best possible, in view of the above remarks. We conclude this section with two examples illustrating what can happen in this respect for other simple equations.

Example 1. Take V to be the affine cone with equation x ² + y ² = z ² . It is easy to see (using e.g. the well known formulae recalled below), that V satisfies assumption VW of Theorem 2 below: namely, roughly speaking there are integral points satisfying any compatible congruence condition.

Also, V is normal and not isomorphic to affine space. Nevertheless, the formulas x = 2abc, y = (a ² −b ² )c, z = (a ² +b ² )c and the similar one obtained by interchanging x, y, produce all the integral solutions by letting a, b, c run through Z. This shows that we cannot in general expect results like Theorem 1 (or Theorem 2) allowing more variables than the dimension. In this example V is not smooth, however. I do not know to what extent the existence of singularities can influence the conclusions in the general case.

Example 2. In contrast with Example 1, we sketch a proof that, for the variety V defined by z ² = 1 + xy, its points over Z cannot be obtained from any number of parametrizations defined over Q ( ¹ ), even if we allow three variables. The argument (like Remark 3!) is related to the lack of units of Z.

(

) This restriction can be eliminated.

Assume the contrary and let φ := (φ 1 , φ 2 , φ 3 ) be one of the relevant parametrizations, where the φ _i ∈ Q[t ₁ , t ₂ , t ₃ ]. Considering the equation (φ ₃ − 1)(φ ₃ + 1) = φ ₁ φ ₂ we easily see that, for some α, β, δ, η ∈ Q[t ₁ , t ₂ , t ₃ ] such that αη − βδ = 2, we have φ 1 = αδ, φ 2 = βη, φ 3 = βδ + 1 ( ² ). Choose now rational integers r, s, d, e with re − sd = 2. Then (rd, se, sd + 1) lies on V(Z), so it is in φ(Z) for some φ as above, taken from a finite set. Comparing with the above equations it is now easy to see that, for some integer a, taken from a finite set ( ³ ), (r, s, d, e) lies in the image on Z ³ of (aα, aβ, a ⁻¹ δ, a ⁻¹ η).

So all the integer solutions of xy − zw = 2 could be obtained from a finite number of polynomial parametrizations in three variables, in contradiction with the obvious analogue of Theorem 1 (which follows in the same way from Theorem 2, and has not been stated for simplicity).

Proofs. Before stating Theorem 2 we introduce some notation and a couple of definitions. For k a number field, we let Σ _k be the set of its places.

For v ∈ Σ k , k v will denote, as usual, the completion of k with respect to v.

We will denote with O _v = O _k,v the valuation ring of k _v . For an affine variety V ⊂ A ⁿ and a ring R, we shall denote by V(R) the set of points of V with coordinates in R.

Definition 1. Let V be an affine variety defined over a number field k.

We say that V has the weak approximation property for a finite set S of places if V(O _k ) is dense in Q

v∈S V(O _v ). We say that V has the very weak approximation property (denoted by VW) if there exists a finite set S ₀ of places of k such that V has the weak approximation property for every finite set S disjoint from S ₀ .

This definition is essentially copied from Def. 3.5.6, p. 29 of [Se], which however refers to projective (instead of affine) varieties and rational (instead of integral) points. From [Se], p. 19, we recall another definition.

Definition 2. A subset A ⊂ V(k) is of type (C1) if it is not Zariski dense in V, of type (C2) if there is a variety V ⁰ with dim V ⁰ = dim V and a generically surjective morphism π : V ⁰ → V of degree ≥ 2 with A ⊂ π(V ⁰ (k)).

Finally, A is called thin if it is contained in a finite union of sets of type (C1) or (C2).

We have the following result (which could be stated in a more general form with A ^d replaced by another d-dimensional affine variety).

Theorem 2. Let V ⊂ A ⁿ be an affine normal variety of dimension d, defined over k and satisfying VW. Let Φ ₁ , . . . , Φ _h : A ^d → V be morphisms.

(

) E.g. put δ := gcd(φ

, φ

− 1) etc.

(

) Here we use the fact that Z has finitely many units.

Suppose that V(O k ) is contained in the union of Φ j (O _k ^d ), 1 ≤ j ≤ h. Then some Φ _j is an isomorphism defined over k.

Note that normality is relevant: without that assumption, the data k = Q, V = {(x, y) : y ² = x ³ }, Φ ₁ : A ¹ → V, Φ ₁ (t) = (t ² , t ³ ), provide a coun- terexample, as is easy to verify.

We let Q be the affine quadric defined by (1). It is a nonsingular, whence normal, variety. Hence Theorem 1 follows at once from Theorem 2 and the next two lemmas. The content of Lemma 1 is certainly known, but, lacking a reference, we give the very short proof.

Lemma 1. Q satisfies VW.

P r o o f. We take S 0 to be the set of archimedean places of k. Then, in view of the Chinese theorem, it plainly suffices to prove the follow- ing: Let a, b, c, d, M ∈ O _k and let ab − cd ≡ 1 (mod M ). Then there exists (a ^∗ , b ^∗ , c ^∗ , d ^∗ ) ∈ Q(O k ) such that (a ^∗ , b ^∗ , c ^∗ , d ^∗ ) ≡ (a, b, c, d) (mod M ). To see this we first observe that (a, c, M )O _k = 1, so, replacing if necessary a with a + tM (for a suitable t ∈ O _k ), we may in fact assume (a, c)O _k = 1.

Put ab − cd = 1 + qM , q ∈ O k and find x, y ∈ O k with ax − cy = q. Then a(b − xM ) − c(d − yM ) = 1, proving what is needed.

Lemma 2. The affine quadric Q defined by (1) is not isomorphic to affine 3-space, even over C.

This is probably well known (in any dimension, for affine quadrics equiv- alent to P _n

i=1 x ² _i = 1), but I have no reference for a direct proof. Here are two simple arguments.

Assume the contrary. Then there exists a morphism f : A ³ → Q such that its inverse g is a regular function on Q, namely given by polynomials in the coordinates x ₁ , . . . , x ₄ . Now, this amounts to the fact that certain equations in the coefficients of the relevant polynomials have a complex solution. By the Nullstellensatz, these equations have already a solution over some number field k. If p is a large prime number and π is a prime ideal in O _k lying above p, we can reduce the coefficients modulo π, producing an isomorphism of Q with affine space, defined over some finite field F q , say.

Now we use a suggestion of D. Zagier: the existence of that isomorphism would imply in particular that the two relevant varieties have the same number of points over F _q , so it suffices to prove this is not true (we are using the zeta function as an invariant). Formulas can be computed at once in our case. Affine 3-space has q ³ points over F q . As to Q, we have plainly

#Q(F _q ) = #SL ₂ (F _q ) = 1

q − 1 #GL ₂ (F _q ) = (q ² − 1)q,

proving what is needed.

For the second argument (which will be useful in connection with Re- mark 2), observe first that by a linear transformation with complex coeffi- cients the equation for Q may be brought in the form x ² ₁ + . . . + x ² ₄ = 1.

Consider the differential form on Q

ω := dx ₁ ∧ dx ₂ ∧ dx ₃

x ₄ .

Observe that ω is regular throughout on Q. In fact, differentiating the defining equation we have x ₁ dx ₁ + . . . + x ₄ dx ₄ = 0 whence, for all j, ω = ±(1/x j ) V

i6=j dx i . Since Q is plainly covered by its intersection with the sets where x _j 6= 0 we get the assertion.

Since ω has maximal dimension (as a holomorphic form), it is certainly closed. If on the other hand it were exact, its integral over the real part of Q, namely over the sphere S ³ , would be zero, in view of the Stokes theorem;

we shall now show that this is not the case. In particular this will prove that Q is not homeomorphic to affine 3-space. Also, the argument automatically proves that S ³ has nontrivial homology class in H 3 (Q) ( ⁴ ).

We can transform ω by using the rational parametrization of Q obtained by intersecting it with the pencil of lines through the point P := (0, 0, 0, 1).

The parametrization takes the form x _i = − 2w _i

σ + 1 for i < 4, x ₄ = σ − 1 σ + 1

where σ := w ₁ ² + w ² ₂ + w ² ₃ . The inverse, defined on Q \ {P }, is given by w _i = x _i /(x ₄ − 1). In particular, this provides a 1-1 C ^∞ map from R ³ to S ³ \ {P }, which can be used for the computation of the above mentioned integral. We find, after some computations,

dx ₁ ∧ dx ₂ ∧ dx ₃ = x ₄

 2

σ + 1

 ₃

dw ₁ ∧ dw ₂ ∧ dw ₃ . Using the above parametrization we therefore find

\

S

ω = \

R

 2

σ + 1

 ₃

dw ₁ ∧ dw ₂ ∧ dw ₃ 6= 0

as wanted ( ⁵ ).

Before proving Theorem 2 we need another lemma.

(

) This fact in turn implies that e.g. there is no morphism φ : A

→ Q which is the identity on Q: otherwise Q would be a retract of affine 4-space, whence would have trivial homology. Of course the existence of such a morphism would produce a parametrization of all integral points.

(

) It seems not obvious how to prove algebraically that ω is not exact.

Lemma 3. Let A be a thin subset of V(k) and let S 0 be a finite set of places of k. Then there exists a finite set S ⊂ Σ _k , disjoint from S ₀ , such that the image of A in Q

v∈S V(O _v ) is not dense.

This is Theorem 3.5.3, p. 28 of [Se], except that k v has been replaced by O _v . Actually, the proof described in [Se], through Propositions 3.5.1 and 3.5.2, gives the present statement.

P r o o f o f T h e o r e m 2. Renumbering indices if necessary, we may assume that Φ ₁ , . . . , Φ _r are defined over k and generically surjective, while, for each j > r, either Φ _j is not defined over k or its image is contained in some proper closed subvariety of Q. Let Φ := (φ 1 , . . . , φ n ) be some Φ j , not defined over k. Here the φ _i are polynomials in d variables, not all of whose coefficients lie in k. So we see that {x ∈ k ^d : Φ(x) ∈ k ⁿ } is contained in a proper algebraic subset of A ^d . In particular Φ(O _k ^d )∩V(O k ) is contained in an algebraic subset of V of dimension ≤ d − 1. Hence there exists an algebraic subset W ⊂ V, of dimension < d, such that

(2) (V \ W)(O _k ) ⊂ [

j≤r

Φ _j (O ^d _k ).

Renumbering again the first r indices if necessary we may assume that Φ ₁ , . . . , Φ _s are birational isomorphisms, while Φ _j has degree ≥ 2 for all j with s < j ≤ r. By adding to W a proper closed subset of V if necessary, we may assume that, for j = 1, . . . , s, the inverse of Φ _j (a rational function on V) is defined on V \ W, so, in particular, each point on (V \ W)(C) is the image under Φ j of a unique point on A ^d (C).

In view of Definition 2 the set A := S

s<j≤r Φ j (O _k ^d ) ∪ W(O k ) is thin, and we have, by (2)

(3) V(O k ) \ A ⊂ [

j≤s

Φ j (O ^d _k ).

If some Φ j , j ≤ s, is an isomorphism we are done, so assume the contrary to derive a contradiction. Let t ₁ , . . . , t _d be the coordinate functions on A ^d . Under each of the Φ ⁻¹ _j , j ≤ s, we may view the t _i as rational functions on V. Since no such Φ j is an isomorphism, for each j ≤ s we may find i = i j

such that t _i lies in k(V) \ k[V]. We denote by g _j such a function. Since V is normal, the only rational functions on Q with trivial divisor of poles are the regular ones (by e.g. [Hart, Prop. 6.3A, p. 132]) and we can thus write, for a suitable prime divisor D _j on V (say defined over k),

g _j = ξ _j /η _j , 1 ≤ j ≤ s,

where ξ j , η j ∈ k[V], ξ j is a unit at D j while η j has positive order at D j .

Denote by P j the (d − 1)-dimensional quasi-affine variety obtained by

removing from D _j the set of zeros of ξ _j . Let x _j be a point on P _j having co-

ordinates which are algebraic over k and let L be the number field generated over k by all such coordinates, with ring of integers O _L .

Also, let S 0 be a finite set of places of k such that V(O k ) is dense in Q

v∈S V(O _v ) for all finite S ⊂ Σ _k disjoint from S ₀ . Select now distinct prime numbers l ₁ , . . . , l _s splitting completely in L and not lying below places in S ₀ , and choose places µ 1 , . . . , µ s ⊂ Σ L , lying resp. above l 1 , . . . , l s . By choosing the l _j large we may assume that the coefficients of the Φ _j and the coordinates of the x _j are µ _j -integers. We have L _µ

= Q _l

for each j ≤ s. Finally, let S 1 := {v 1 , . . . , v s } be the set of places of k lying resp. below µ 1 , . . . , µ s . Plainly O _L,µ

= O _k,v

= Z _l

.

By Lemma 3 we may find a set S 2 ⊂ Σ k , disjoint from S 0 ∪ S 1 , such that the image of A in Q

v∈S

V(O _v ) is not dense. Let Ω be an open subset of Q

v∈S

V(O _v ) disjoint from the image of A.

Observe that, for each j ≤ s, there exists a neighborhood I j of x j in V(O _L,µ

) (we remark that x _j ∈ V(O _L,µ

)) such that, for x ∈ I _j , g _j (x) is not a µ j -integer whenever it is defined: this is true because η j (x j ) = 0, while ξ _j (x _j ) 6= 0. Since O _L,µ

= O _k,v

= O _v

, I _j is a neighborhood of x _j in V(O _v

).

By the defining property of S ₀ we may find a point α ∈ V(O _k ) such that its image in Q

v∈S

∪S

V(O v ) lies in ( Q

j≤s I j ) × Ω. We contend that any such point provides the desired contradiction. In fact, first of all, α cannot lie in A, by our choice of Ω. In particular, α cannot lie on W, so Φ ⁻¹ _j is defined at α for j ≤ s and we have α = Φ _j (β _j ), for a uniquely determined β _j ∈ A ^d . Now, g _j (α) is equal to some coordinate of β _j and is not a v _j -integer. Hence α cannot be in the image of Φ j on O ^d _k and, since α does not lie in A, we have a contradiction with (3).

R e m a r k 1. It is possible to prove Theorem 1 in an ad hoc manner, using the ordinary version of the Hilbert Irreducibility Theorem (see e.g.

[La], [Sch] or [Se]) in place of Lemma 3. (In fact, the concept of very-weak approximation and the above Lemma 3 are related to Hilbert’s theorem and indeed lead to a version of it; this is due to J.-L. Colliot-Th´el`ene [CT] and T. Ekedahl [Ek]; see [Se, Thm. 3.5.7, p. 30] and [Se, Prop. 3.3.1, p. 23]).

We may roughly describe such a proof as consisting of two parts. First

a rationality part, based on H.I.T.: we forget integrality conditions and

simply show that, for certain points α on V, the corresponding possible

values of the coordinates t i on A ^d , for points in the inverse image relative to

some Φ _j , cannot be even rational (this happens when deg Φ _j ≥ 2). Second,

an integrality part, equal to the above: we show that those values of the

parameters which could be possibly rational, cannot in fact all be integral

(this has to do with the Φ _j having degree 1). The first part is related to

Hilbert’s theorem, while the argument for the second one consists essentially

in a crude form of Weil’s Decomposition Theorem (see [We] or [La, p. 263]),

stating roughly that the factorization of the divisor of a rational function implies a corresponding factorization of its values at algebraic points.

R e m a r k 2. It is possible to prove a sharper version of Theorem 1;

namely, even letting the variables t i run through the ring O of all algebraic integers in place of O _k , we cannot obtain all solutions in O _k ; this sharper result requires a corresponding sharpening of the above Lemma 2. Namely we need the following statement: there does not exist a finite morphism from affine 3-space to Q. A proof of this result may be obtained as in our second argument for the present Lemma 2: if such a morphism existed the form ω would induce a regular closed form ω ^∗ on A ³ , which would be exact. Now, taking the trace of an equation ω ^∗ = dη, say, we expect to show that ω itself would be exact, a fact which we have proved to be false. However, that the trace sends regular forms into regular ones seems to me not so automatic.

I have now found a proof, written in a forthcoming paper [Za]; in fact, I have found no references for the needed result. Alternatively, C. Deninger has pointed out to me that a proof of the improved lemma may be ob- tained (in any characteristic 6= 2) using ´etale cohomology of quadrics and the pushforward map.

To derive the above contention from the sharpened Lemma 2, we may prove an analogue of Theorem 2, assuming now that V(O _k ) ⊂ S

j≤h Φ _j (O ^d ), to conclude that some Φ j is a finite morphism defined over k. The proof runs like the above one, but now it is necessary to consider, in addition to the g _j , the rational functions on V obtained as follows. Let Φ _j be a generically surjective one among the given morphisms. Then we may view the function field of A ^d as a finite extension of the function field of V. So the coordinate functions t _i on A ^d satisfy monic irreducible equations over k(V). By [Se], Prop. 3.3.1, p. 23, such equations remain irreducible over k if we specialize the coefficients to points in V(O _k ), except for a thin set. If for given j all the coefficients of such equations are regular functions on V, we see that Φ j is finite and we are done. Otherwise some such coefficient will have nontrivial divisor of poles. We must take into account precisely the set of such coefficients (which includes the set of the g j , considered in the above proof). The rest of the argument is exactly the same as above.

R e m a r k 3. As announced in the introduction we shall sketch (condi- tional) arguments which show that the analogue of Theorem 1 is hardly true if five variables are allowed.

We start with the parametrization of Q coming from the general simple continued fraction with five partial quotients, namely the fraction

a ₀ + 1 a 1 +

1 a 2 +

1 a 3 +

1 a 4

.

If p 3 /q 3 and p 4 /q 4 are the last two convergents, we see, putting x 1 = p 3 , x 2 = q ₄ , x ₃ = p ₄ , x ₄ = q ₃ , that x ₁ , x ₂ , x ₃ , x ₄ are polynomials in a ₀ , . . . , a ₄ satisfy- ing (1). Also, we find that, identically,

(4)

a 0 = (1 + a 3 a 4 )x 1 − a 3 x 3

(1 + a ₃ a ₄ )x ₄ − a ₃ x ₂ , a 1 = (1 + a 3 a 4 )x 4 − a 3 x 2 , a ₂ = x ₂ − a ₄ x ₄ − 1

(1 + a 3 a 4 )x 4 − a 3 x 2

.

Let now P ^∗ := (x ^∗ ₁ , x ^∗ ₂ , x ^∗ ₃ , x ^∗ ₄ ) ∈ Q(O _k ). We substitute x ^∗ _i for x _i in (4) and try to choose a ₃ , a ₄ ∈ O _k to make a ₁ = (1 + a ₃ a ₄ )x ^∗ ₄ − a ₃ x ^∗ ₂ a unit u ∈ O ^∗ _k . If this may be done, then even a 0 , a 2 will lie in O k , so the point P ^∗ will be obtained integrally from the parametrization.

Our equation is u = (1 + a ₃ a ₄ )x ^∗ ₄ − a ₃ x ^∗ ₂ = x ^∗ ₄ + a ₃ (a ₄ x ^∗ ₄ − x ^∗ ₂ ). Hence it suffices if a 4 ∈ O k may be found with x ^∗ ₄ ≡ u (mod x ^∗ ₂ − a 4 x ^∗ ₄ ). Our problem will be solved provided it is possible to choose a ₄ ∈ O _k such that π := x ^∗ ₂ − a 4 x ^∗ ₄ is a prime element in O k with the property that the cyclic group (O _k /(π)) ^∗ is in fact equal to the reduction (mod π) of the unit group:

this condition is plainly sufficient for the above congruence to be solvable in u.

To fix ideas, let k = Q( √

2), so O _k = Z[ √

2]. Now each unit is of the form ±ε ^m , where ε = 1 + √

2. To reach the above goal we must first produce

prime elements π ≡ x ^∗ ₂ (mod x ^∗ ₄ ). This can be achieved by the generalized

Dirichlet theorem, since (x ^∗ ₂ , x ^∗ ₄ ) = 1 for any P ^∗ ∈ Q(O _k ), but we have

the much stronger restriction that ε has to be an (almost) primitive root

mod π. We are faced with an analogue of Artin’s well known conjecture

for primitive roots. Even in the case of the rational field this remains un-

solved; however, C. Hooley [Ho] has proved that, if the generalized Riemann

conjecture is true for Dedekind zeta functions of suitable fields, there are

primes π ∈ N (and actually they have the “right” density) for which a given

nonsquare integer e 6= −1 is a primitive root. Hooley starts by observing

that e is a primitive root mod π iff for any prime l | π − 1 the congruence

x ^l ≡ e (mod π) is not solvable. Reversing the procedure, he then starts

from all such congruences and sieves out the bad primes. Hooley’s method

has been followed by J. P. Weinberger [Wei], who showed (again under Rie-

mann hypothesis) that UFD is euclidean, for rings of integers of number

fields containing nontrivial units. He proves our contention when x ^∗ ₄ is a

prime element, but his argument works generally. One proceeds as in [Ho],

but considering only primes π in the generalized progression x ^∗ ₂ mod x ^∗ ₄

(of course associate primes give rise to the same condition). Again one is

led to the congruences x ^l ≡ ε (mod π), where l divides |N _Q ^k (π)| − 1. These

congruences represent the essentially new part, compared to Hooley: for the

existence of primes π with the required properties it is of course necessary

that each such congruence is not solvable for some of the primes in the rele- vant progression, a fact which is not so obvious, since lying in the progression could possibly influence the solvability (e.g. x ² ≡ 5 (mod p) is solvable for all primes p = 5m±1). This is not so in our case and a proof may be found in Vol. II of H. Hasse’s Bericht ¨uber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlk¨orper, 1930, reprinted 1965. Of course one needs quantitative results on the densities and remainder terms to carry out the above procedure, and it is here that Riemann’s conjecture is needed.

We conclude this remark by recalling that partial, but unconditional, results towards Artin’s conjecture were obtained by R. Gupta and M. Ram Murty [G-RM] and D. R. Heath-Brown [HB] for the classical case k = Q and by W. Narkiewicz for certain cyclotomic fields k. To my knowledge, however, no generalization of the original Gupta–Ram Murty method to number fields is already available for our application, even when k has large degree.

R e m a r k 4. As announced before, we describe a possibility for obtain- ing an example over Z, starting from the one coming from Remark 3, and formulate a general problem on polynomial ideals which arises in this con- nection.

We assume (as in Remark 3) the existence of polynomials P ₁ , . . . , P ₄

∈ Z[t 1 , . . . , t 5 ], with P 1 P 2 − P 3 P 4 = 1 and such that, for each solution (a ₁ , . . . , a ₄ ) ∈ Z[ √

2] ⁴ of (1), there exist b ₁ , . . . , b ₅ ∈ Z[ √

2] such that P i (b 1 , . . . , b 5 ) = a i for 1 ≤ i ≤ 4.

Write t _i = u _i + √

2v _i , for indeterminates u _i , v _i . Then we may write P _i (t ₁ , . . . , t ₅ ) = Q _i + √

2R _i

for polynomials Q i , R i ∈ Z[u 1 , . . . , u 5 , v 1 , . . . , v 5 ]. Now assume (a 1 , . . . , a 4 ) ∈ Z ⁴ is a solution of (1) in rational integers, and pick b ₁ , . . . , b ₅ as above. We may put b _i = u ^∗ _i + p

v _i ^∗ for rational integers u ^∗ _i , v _i ^∗ and consequently we get, in an obvious notation,

(5) Q i (u ^∗ , v ^∗ ) = a i , R i (u ^∗ , v ^∗ ) = 0

for all i. Define R := Z[u 1 , . . . , u 5 , v 1 , . . . , v 5 ]. Now we make the following Assumption. There exist polynomials F, G in the ideal I generated by R ₁ , . . . , R ₄ in R, such that Q ₁ + F, Q ₃ + G generate the unit ideal in R.

Observe that the assumption is certainly satisfied if we replace the ring R with the ring R[ √

2] = Z[ √

2][u ₁ , . . . , u ₅ , v ₁ , . . . , v ₅ ] (we may take F =

√ 2R ₁ , G = √

2R ₃ ). Also, the polynomials P _i coming from Remark 3 may

be easily explicitly computed, so it may be that F , G can be found without

appealing to any general assertion leading to the above assumption. How-

ever, we have not succeeded in doing so at the moment, nor to disprove the

assumption for the polynomials in question.

Let then F, G be as in the assumption, so we have an equation

(6) X(Q ₁ + F ) + Y (Q ₃ + G) = 1

for suitable X, Y ∈ Z[u ₁ , . . . , u ₅ , v ₁ , . . . , v ₅ ].

Since P 1 , . . . , P 4 satisfy (1) we have in particular (7) Q ₁ Q ₂ − Q ₃ Q ₄ + 2R ₁ R ₂ − 2R ₃ R ₄ = 1.

Multiply both terms of (6) by ∆ := 2R ₁ R ₂ − 2R ₃ R ₄ − F Q ₂ + GQ ₄ ∈ I. We get, using (7),

(∆X)(Q ₁ + F ) − (∆Y )(Q ₃ + G) = 2R ₁ R ₂ − 2R ₃ R ₄ − F Q ₂ + GQ ₄

= 1 − Q ₁ Q ₂ + Q ₃ Q ₄ − F Q ₂ + GQ ₄ , which may be rewritten as

(Q ₁ + F )(Q ₂ + ∆X) − (Q ₃ + G)(Q ₄ + ∆Y ) = 1,

namely the four polynomials S ₁ := Q ₁ + F, S ₂ := Q ₂ + ∆X, S ₃ := Q ₃ + G, S 4 := Q 4 + ∆Y give a solution of (1) and lie in Z[u 1 , . . . , u 5 , v 1 , . . . , v 5 ].

Take now a solution a ₁ , . . . , a ₄ of (1) in rational integers, and find rational integers u ^∗ _i , v ^∗ _i as above, so (5) is verified. From (5) we see that, for every polynomial Γ ∈ I, we have Γ (u ^∗ ₁ , . . . , u ^∗ ₅ , v ₁ ^∗ , . . . , v ₅ ^∗ ) = 0. Since F, G, ∆ ∈ I we finally obtain, using again (5),

S _i (u ^∗ ₁ , . . . , u ^∗ ₅ , v ₁ ^∗ , . . . , v ₅ ^∗ ) = a _i , i = 1, 2, 3, 4, so the polynomials S i would produce the desired example over Z.

We conclude by mentioning a simply stated problem on polynomial ide- als, somewhat relevant in connection with the above “Assumption”:

Problem. Let a, b, c ∈ Ω := k[x ₁ , . . . , x _n ] generate the unit ideal. Is it necessarily true that there always exist u, v ∈ Ω such that a + uc, b + vc generate the unit ideal?

At first sight I would expect a negative answer, but have no counterex- ample. We omit natural generalizations and modifications of the problem (e.g. does the answer depend on k, or can one find a decision algorithm, given a, b, c?), also in view of the fact that they fall too far from the main topic of the paper.

References

[CT] J.-L. C o l l i o t - T h´el`en e, Letter to T. Ekedahl, 9/21/1988, unpublished.

[Ek] T. E k e d a h l, An effective version of Hilbert’s Irreducibility Theorem, preprint,

Stockholm 1987 (see also S´eminaire de Th´eorie des Nombres, Paris 1988–1989,

Birkh¨auser, 1990, 241–248).

[G-RM] R. G u p t a and M. R a m M u r t y, A remark on Artin’s conjecture, Invent. Math.

78 (1984), 127–130.

[Hart] R. H a r t s h o r n e, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.

[HB] D. R. H e a t h - B r o w n, Artin’s conjecture for primitive roots, Quart. J. Math.

Oxford 37 (1986), 27–38.

[Ho] C. H o o l e y, On Artin’s conjecture, J. Reine Angew. Math. 225 (1967), 209–220.

[La] S. L a n g, Fundamentals of Diophantine Geometry, Springer, 1983.

[Sch] A. S c h i n z e l, Selected Topics on Polynomials, University of Michigan Press, Ann Arbor, 1982.

[Se] J. P. S e r r e, Topics in Galois Theory, Jones and Bartlett Publ., Boston, 1992.

[Sko] T. S k o l e m, Diophantische Gleichungen, Springer, 1938, reprint Chelsea, 1950.

[Sz] K. S z y m i c z e k, On some diophantine equations connected with triangular num- bers, Zeszyty Nauk. Wyż. Szkoły Ped. w Katowicach Mat. 4 (1964), 17–22 (in Polish).

[We] A. W e i l, L’arithm´etique sur les courbes alg´ebriques, Acta Math. 52 (1928), 281–

315 = Coll. Papers, Vol. I, 2nd printing, Springer, 1980, 11–46.

[Wei] P. J. W e i n b e r g e r, On Euclidean rings of algebraic integers, in: Proc. Sympos.

Pure Math. 24, Amer. Math. Soc., 1973, 321–332.

[Za] U. Z a n n i e r, Note on traces of regular differential forms, preprint, 1995.

Ist. Univ. Arch. D.S.T.R.

S. Croce, 191 30135 Venezia, Italy

E-mail: zannier@dimi.uniud.it

Received on 12.2.1996

and in revised form on 2.7.1996 (2927)